Algebraic Operads (version 0.99, draft 2010) by Jean-Louis Loday, Bruno Vallette

By Jean-Louis Loday, Bruno Vallette

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Example text

Equipped with this symmetry gVect is simply called the category of graded vector spaces. 3. Koszul sign rule. When working in the symmetric monoidal category of sign-graded vector spaces with switching map τ , there are signs involved in the formulas. For instance, if A is a graded algebra, then the product in A ⊗ A is given by (x ⊗ y)(x ⊗ y ) = (−1)|y||x | (xx ) ⊗ (yy ). In order to avoid complicated signs in formulas, in particular signs depending on the degree of the elements involved, there is a useful convention, called the Koszul sign rule, which is as follows.

For instance, the first differential d0 is given by the part of the differential d which lives in Fp and not in Fp−1 . Throughout this book, we will encounter many chain complexes with differential maps made up of the sum of several terms. So the aforementioned theorem will be the main tool to investigate their homology. 8. Differential graded algebra. A graded algebra A is a graded vector space {An }n≥0 equipped with a unital product µ of degree 0. Hence it sends Ap ⊗Aq into Ap+q . For instance, if we put V in degree 1, then the tensor algebra T (V ) is a graded algebra (cf.

The category gVect, equipped with this symmetry, is called the category of sign-graded vector spaces. Observe that there is another choice of symmetry: τ (v ⊗w) := w ⊗v. Equipped with this symmetry gVect is simply called the category of graded vector spaces. 3. Koszul sign rule. When working in the symmetric monoidal category of sign-graded vector spaces with switching map τ , there are signs involved in the formulas. For instance, if A is a graded algebra, then the product in A ⊗ A is given by (x ⊗ y)(x ⊗ y ) = (−1)|y||x | (xx ) ⊗ (yy ).

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